Effective apparent power definition based on sequence components for non-sinusoidal electric power quantities

Electric Power Systems Research 117 (2014) 210–218

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Effective apparent power definition based on sequence components for non-sinusoidal electric power quantities O. Boudebbouz ∗ , A. Boukadoum 1 , S. Leulmi 2 Department of Electrical Engineering, Laboratoire d’électrotechnique de Skikda “LES”, Université du 20 Août 1955, Skikda 21000, Algeria

a r t i c l e

i n f o

Article history: Received 10 July 2012 Received in revised form 8 February 2014 Accepted 22 August 2014 Available online 16 September 2014 Keywords: Effective apparent power Symmetrical harmonic components Power quality Unbalance Harmonics Non-sinusoidal systems

a b s t r a c t In the present paper, effective apparent power based sequence components for non-sinusoidal electric quantities is proposed. The effective voltages and currents based sequence components contained in the old and new versions of the IEEE Std 1459 are given, only, at the fundamental frequency. These effective quantities are derived for the non-sinusoidal situations by means of the symmetrical harmonic components. The latter are obtained by the use of three transformation matrices. Two tests systems are used for the validation process. The resulted effective quantities based on symmetrical harmonic components are very similar to those based on the phase reference frame. The new alternative expressions can be considered as a new vision of power quality research and update, as well as, for diagnosis purposes in distorted and unbalanced power systems. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The definition of apparent power for unbalanced or non-sinusoidal three-phase systems is still a controversial issue [1–6]. In the literature, one can find few suggested methods they can be used for unbalanced and non-sinusoidal electric quantities [5,7,8]. In the IEEE approach, effective voltages and currents have been presented as convenient quantities which give more realistic energy flow in distorted and unbalanced circuits. The effective quantities are given for three or four wire systems in phase reference frame. The alternative expressions in sequence reference frame have been given only at the fundamental frequency. This limitation calls for a better method that can give symmetrical harmonic components for overall range of integer harmonic orders. Recently, it has been proven that generalized application of the Fortescue method to harmonic components is not possible. Additional components have been proposed to overcome the limitation of the Fortescue method for non-sinusoidal situations [9]. To the authors’ knowledge, the work in reference [10] related to Girgis et al. was the first who proposed three transformation matrices giving symmetrical harmonic components similar to that of the Fortescue method. Each matrix is applied to a sub-set of harmonic orders of which the corresponding balanced three phase quantities have positive rotating sequence or negative rotating sequence or without rotating sequence corresponding to the zero sequence. The unbalanced voltages and currents phasors abc at hth harmonic order is then transformed to new three phasors called balanced, first unbalanced and second unbalanced. This new method has been presented as a powerful tool for the diagnosis purposes and the resonance occurrence evaluation as well as for modern power theory in distorted and unbalanced power systems [10–12]. More recently, Langella et al. proposes a new unique transformation matrix that is capable of suitably extracting the balanced, first unbalanced, and second unbalanced components suitable for all of the harmonic and interharmonic orders [13]. This method uses a same designation as Girgis method and its transformation matrix has a generic property for all harmonic and interharmonic components and conserves some similarities with the Girgis method. In the present paper, effective voltages and currents contained in the old and new versions of the IEEE Std 1459 are extended to sequence harmonic components by means of the Girgis method. The great merit of the extended formulas to non-sinusoidal situations is that it offers

∗ Corresponding author. Tel.: +213 771 785796. E-mail addresses: [email protected] (O. Boudebbouz), [email protected] (A. Boukadoum), [email protected] (S. Leulmi). 1 Tel.: +213 7 71386368. 2 Tel.: +213 7 76 11 07 31. http://dx.doi.org/10.1016/j.epsr.2014.08.017 0378-7796/© 2014 Elsevier B.V. All rights reserved.

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211

the assessment of unbalance in the presence of harmonics which can interest some power quality objectives by considering new indices. Two examples are used to verify the validity of the proposed formulas. 2. Symmetrical harmonic components It is known that for the perfectly balanced and distorted three-phase electric circuits, the voltages (or currents) of order 1, 4, 7, 10, 13,. . . correspond to a positive sequence. The voltages (or currents) of order 2, 5, 8, 11, 14,. . . correspond to a negative sequence. The voltages (or currents) of order h = 3, 6, 9, 12, 15,. . . correspond to a zero sequence. They are designed as subsets G1 , G2 and G0 , respectively. A new decomposition method has been proposed which gives the symmetrical harmonic components corresponding to the overall range of harmonic orders [10]. A new more coherent designation takes place defining, at each harmonic order, the balanced components (bn), the first unbalanced components (fu) and the second unbalanced components (su) resulting from the application of the matrices (1)–(3):

⎡

a

a2

TG1 = 1/3 ⎣ 1 a2

a

1

1

1 a2

a

1

⎢

1

⎡ ⎢

⎤

⎥ ⎦ ; h = 3m − 2 ⎤ ⎥

TG2 = 1/3 ⎣ 1

a

a2 ⎦ ;

1

1

1

⎡

1

1

⎢ ⎢1 ⎣ 1

h = 3m − 1

(2)

⎤

1

√ −1 − 3 2 √ −1 + 3 2

TG0 = 1/3 ⎢

(1)

√ −1 + 3 2 √ −1 − 3 2

⎥ ⎥ ⎥ ; h = 3m ⎦

(3)

where h corresponds to the hth harmonic order, the integer number m ∈ N* and the complex operator a = e(j2)/3 . TG1 corresponds to a sub-set G1 . TG2 corresponds to a sub-set G2 . TG0 corresponds to a sub-set G0 . The matrices show evidence of the presence of the unbalance at each harmonic order including the fundamental one. The phasors of three-phase voltages (or currents) for each harmonic order can be found by:

⎡ ¯h ⎤ ⎡ ¯h⎤ Va Vbn ⎢ ¯h ⎥ ⎢ ¯h⎥ ⎣ Vfu ⎦ = [M] ⎣ Vb ⎦

(4)

V¯ ch

h V¯ su

[M] corresponds to one of the three matrices TG1 , TG2 and TG0 according to the harmonic order. 3. Effective electric power quantities based sequence components 3.1. IEEE approach Recent proposals of effective electric quantities have been given as more realistic physical meaning of energy flow in three-phase power systems. According to the reference [14], the expression of effective voltage (Ve ) is given for four wires systems as follows:



2 + V2 Ve1 eH

Ve =

(5)

where Ve1 and VeH the fundamental and the non-fundamental effective voltages, respectively:



Ve1 =

VeH =



1 Vab

2



1 + Vbc

H 

h=2

2



1 + Vca

2

+3



Va1

2

18 h Vab

2



h + Vbc

2



h + Vca

2



+ Vb1

+3

2

 2 h Va



+ Vc1

2

(6)

2

+ Vbh

2 

+ Vch

18

(7)

1 , V 1 and V 1 characterize the rms fundamental line to line voltages. Va1 , Vb1 and Vc1 are the rms fundamental line to neutral voltages and Vab ca bc

h , V h and V h denote the rms line to line voltages, both at the Whereas, Vah , Vbh and Vch represent the rms line to neutral voltages and Vab ca bc hth harmonic order. The dc component is, generally, neglected in practical electric systems and the range of harmonic orders has a finite number noted H which, generally, does not exceed 100 [15]. The corresponding effective current:



Ie =

Ia2 + Ib2 + Ic2 + In2 3

Ia , Ib , Ic and In are the rms value of the distorted phase and neutral currents, respectively.

(8)

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The alternative expressions in sequence reference frame for voltages and currents are given, only, at the fundamental frequency [14]. With the new designation at the subscript, they are given as follows:

Ve1 =





1 2 Vbn

+



1 2

Ie1 =

Ibn



2

1 Vfu

 2

1 + Ifu

+

1 Vsu

2 (9)

2

2

1 + 4 Isu

(10)

3.2. Generalized effective electric quantities based symmetrical harmonic components The use of the matrices (1)–(3) allows us to transform distorted and unbalanced three-phase voltages or currents to their corresponding sequence components for any hth harmonic order of interest. Using the two matrices manipulations (11) and (12), the following relationship between phase and sequence rms voltages can be obtained such as shown in (13):

⎡ ¯ h ⎤T ⎛ ⎡ ¯ h ⎤⎞T ⎡ ¯ h ⎤T Va Va Vbn ⎢ ¯h ⎥ ⎜ ⎢ ¯ h ⎥⎟ ⎢ ¯h⎥ T ⎣ Vfu ⎦ = ⎝[M] ⎣ Vb ⎦⎠ = ⎣ Vb ⎦ [M] V¯ ch

h V¯ su

(11)

V¯ ch

⎡ ¯ h ⎤∗ ⎡ ¯ h ⎤∗ Va Vbn ⎢ ¯h ⎥ ∗⎢ ¯ h ⎥ ⎣ Vfu ⎦ = [M] ⎣ Vb ⎦ V¯ ch

h V¯ su



h Vbn

(12)

2



h + Vfu

2



h + Vsu

2

1 3

=

 2 h

2

2

+ Vbh

Va

+ Vch

(13)

‘T’ and ‘*’, at the superscript, represent the transpose and the conjugate, respectively. The sequence components of line-to-line voltages at 3m − 1 and 3m − 2 orders are obtained in the same way. Moreover, they lack the second unbalanced component and have the following properties:

⎡ ¯h ⎤ Vab ⎢ ¯h ⎥ ⎣ Vbc ⎦ h V¯ ca

⎡

1

=⎣

a2 a

G1

⎡ ¯h ⎤ Vab ⎢ ¯h ⎥ ⎣ Vbc ⎦ h V¯ ca

⎡

⎤   h V¯ bnll a⎦

1

a2

G1

h V¯ full

(14) G1

⎤   h V¯ bnll 2 =⎣ a a ⎦ 1

a2

G2

1

a

G2

h V¯ full

(15) G2

It is noted that the line to line voltages squared are determined by multiplying the phasors with its conjugated values and for both the sub-sets G1 and G2 , we can write:



h Vbnll

2



h + Vfull

2

=

1 3



h Vab

2



h + Vbc

2



h + Vca

2

(16)

The sequence components of the line-to-line voltages at triplen harmonics lack the balanced components and, similar to the sub-sets G1 and G2 , we can write:



h Vfull

2



h + Vsull

2

=

1 3



h Vab

2



h + Vbc

2



h + Vca

2

(17)

h , V h and V h are the balanced, the first and the second unbalanced line-to-line voltages at hth harmonic order, respectively. where Vbnll full sull The substitution of (13), (16) and (17) into (5) gives the expression of the generalized effective voltages, in the sequence reference frame, as follows:

 ⎧

⎫    2 h  2 ⎪  H ⎪ h 2 + Vh H ⎨ ⎬ V + V   su bn fu Ve =  +  2 ⎪  h=1 ⎪ ⎩ ⎭ h=1 

⎧

⎫  2  h 2 ⎪ ⎪ h ⎨ Vbnll + Vfull ⎬ ⎪ ⎩

6

⎪ ⎭

⎧  2 ⎫

h 2 ⎪ ⎪ h ⎨ ⎬ V + V  full sull H

+

h=3,6,...

⎪ ⎩

6

⎪ ⎭

(18)

h= / 3, 6, . . . The phase currents based harmonic sequence components are derived in the same way as for the line-to-neutral voltages. Therefore, the effective current expression in (8) requires further developments related to the presence of the neutral current. This latter will be derived by the following analysis.

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The instantaneous phase currents have the following expressions:

⎧ H  ⎪

 √ ⎪ ⎪ i = 2Iah sin hωt + ϕah ⎪ a ⎪ ⎪ ⎪ h=1 ⎪ ⎪ ⎪ H ⎨   √ 2h 2Ibh sin hωt −

ib =

+ ϕbh

 (19)

3 ⎪ ⎪ h=1 ⎪ ⎪ ⎪ H   ⎪ √ ⎪ 2h ⎪ ⎪ ic = 2Ich sin hωt + + ϕch ⎪ ⎩ 3 h=1

The current in the neutral ‘in ’ is derived from: in = ia + ib + ic

(20)

The developed form is shown in (21) in =

H 



H   h



ˆIah sin hωt + ϕah +

h=1

ˆI sin hωt − b

h=1

2h + ϕbh 3

H  



2h + ϕch 3

ˆIch sin hωt +

+

h=1

 (21)

For each harmonic order, the current phasors in phase reference frame versus the corresponding sequence components can be found by the application of the following matrices: TG−1 for h = 3m − 2, TG−1 for h = 3m − 1 and TG−1 for h = 3m. 1

2

Based on (20), the phasors of neutral current at each harmonic order are obtained:

'

h , I¯nh = 3I¯su

h = 3m − 1 or h = 3m − 2

h , I¯nh = 3I¯bn

h = 3m

0

(22)

Based on the system in (22), the distorted neutral current can be transformed into time domain as shown in (23). in = 3

H 





h h ˆIsu sin hωt + su +3

H 



ˆI h sin hωt +  h bn bn



(23)

h=3,6,...

h=1 / 3, 6, . . . h=

The rms value of the neutral current, in the sequence reference frame, can be derived from (23) as follows:

  H   In = 3   h = 1

h 2 Isu

+

H 

h 2

Ibn

(24)

h=3,6,...

h= / 3, 6, . . . Then, the generalized effective current, in the sequence reference frame, is shown in (25).

  H ( H   2  2 2 ) h h h Ie =  I + I + I + 3 su  bn fu  h=1 h=1 

h 2 Isu

+3

H 

h 2

Ibn

(25)

h=3,6,...

h= / 3, 6, . . . 4. Effective apparent power based sequence components Based on (18) and (25), the effective apparent power based symmetrical harmonic components are, easily, found by the relation: Se = 3Ve Ie . The effective harmonic apparent power relation is given by: SeH = 3VeH IeH where VeH and IeH take the same expressions as in (18) and (25) but for h ≥ 2. For h = 1, we have power.  the effective fundamental apparent

h 

h  √ √ √ h h = 3V h h Using Vbnll = 3Vbn , Vfull and V = 3Vsu , a simple rearrangement of (18) and (25) gives more fu sull h∈G1 ,G2

h∈G0

h∈G1 ,G2 ,G0

coherent formulations of effective voltage and current as follows:

'

1 2 *  H    2  2 Vsu 1 1  Ve =  Vbn + Vfu + + 2  h≥2  / 3, 6, . . . h=

'



h 2 Vbn

+



h Vfu

2

+

h Vsu

2

2 * +

H  h=3,6,...

' 

h Vfu

2

+





h 2 Vsu

+

h Vbn

2

2 * (26)

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 ( ) H    2  2

1 2 1 1  Ie =  Ibn + Ifu + 4 Isu +  h≥2 

(

h 2 Ibn

+

 2 h Ifu

h 2

+ 4 Isu

) +

( 2 H  h Ifu

h 2

+ Isu

h 2

+ 4 Ibn

) (27)

h=3,6,...

h= / 3, 6, . . . Eqs. (26) and (27) seem to be more accurate in comparison with forward generalization of (9) and (10) for harmonic domain such as reported in [16]. Eqs. (26) and (27) can be written in condensed form as follows:



(Ve1 )2 + (VeH12 )2 + (VeH0 )2

Ve =

 Ie =

(28)

(Ie1 )2 + (IeH12 )2 + (IeH0 )2

(29)

H12 represents subsets G1 and G2 except the fundamental component. H0 represents a subset G0 . Based on (28) and (29), the effective apparent power will be, so, derived as follows: Se2 = 9Ve2 Ie2 = 9((Ve1 )2 + (VeH12 )2 + (VeH0 )2 ) ((Ie1 )2 + (IeH12 )2 + (IeH0 )2 )

(30)

The new definition of effective apparent power can be divided as: Se2 = (Se1 )2 + (SeN )2

(31)

(Se1 )2 = 9(Ve1 )2 (Ie1 )2

(32)

while

and

+

(SeN )2 = 9 (Ve1 )2 (IeH12 )2 + (Ve1 )2 (IeH0 )2 + (VeH12 2 )2 (Ie1 )2 + (VeH0 )2 (Ie1 )2 + (VeH12 )2 (IeH12 )2 +(VeH0 )2 (IeH12 )2 + (VeH12 )2 (IeH0 )2 + (VeH0 )2 (IeH0 )2

,

(33)

9 (Ve1 )2 (IeH12 )2 , 9 (Ve1 )2 (IeH0 )2 , 9 (Ie1 )2 (VeH12 )2 and 9(Ie1 )2 (VeH0 )2 the four major terms that contribute to SeN . They are, mainly, due to fundamental effective quantities. 9(VeH12 )2 (IeH12 )2 , 9(VeH0 )2 (IeH12 )2 , 9(VeH12 )2 (IeH0 )2 and 9(VeH0 )2 (IeH0 )2 are harmonic effective apparent powers which are net product of harmonic effective quantities. It is important to note that each term of Se contains sequence harmonic components so we can write for fundamental frequency:

(Se1 )2 = 9

⎧ 1 )2 (I 1 )2 + (V 1 )2 (I 1 )2 + 4(V 1 )2 (I 1 )2 ⎪ (Vbn su ⎪ bn bn fu bn ⎪ ⎨ 2

2

2

2

2

1 ) (I 1 ) + (V 1 ) (I 1 ) + 4(V 1 ) (I 1 ) (Vfu su bn fu fu fu

2

⎫ ⎪ ⎪ ⎪ ⎬

(34)

⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (V 1 )2 (I 1 )2 + 1 (V 1 )2 (I 1 )2 + 2(V 1 )2 (I 1 )2 ⎪ ⎭ su su su su bn fu 2

2

If only, one harmonic order of subsets G1 or G2 is considered, we have:

9(VeH12 )2 (IeH12 )2 = 9

⎧ h )2 (I h )2 + (V h )2 (I h )2 + 4(V h )2 (I h )2 ⎪ (Vbn su ⎪ bn bn fu bn ⎪ ⎨ 2

2

2

2

2

h ) (V h ) (I h ) + (V h ) (I h ) + 4(V h ) (Isu

fu bn fu fu fu ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (V h )2 (I h )2 + 1 (V h )2 (I h )2 + 2(V h )2 (I h )2 ⎪ ⎭

2

su

bn

su

2

su

fu

⎧ ⎪ ⎪ ⎪ ⎨

2

2

2

2

2

2

2

2

2

2

2

2

h ) (I h ) + (V h ) (I h ) + 4(V h ) (I h ) (Vfu su fu fu fu bn

h ) (I h ) + (V h ) (I h ) + 4(V h ) (I h ) (Vsu su su su

⎫ ⎪ ⎪ ⎪ ⎬

fu bn ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (V h )2 (I h )2 + 1 (V h )2 (I h )2 + 2(V h )2 (I h )2 ⎪ ⎭ su bn fu bn bn bn

2

(35)

su

and if only, one harmonic order of subsets G0 is considered, we have:

9(VeH0 )2 (IeH0 )2 = 9

2

⎫ ⎪ ⎪ ⎪ ⎬

(36)

2

Cross frequency terms can be derived in the similar manner. If one takes into account overall range of harmonic orders, a large calculation time is required. For example, for the fundamental frequency and, only, two harmonic orders of interest, we have for Se1 , 9 computational terms and 72 computational terms for SeN . In the general case, have orders of interest, we have for SeN , (3n × 3n) − 9

1 when1 we

n harmonic  1 1 cos  1 computational terms. All of the power terms other than Pbn Pbn = 3Vbn Ibn are non-efficient power because they are not converted bn to useful work and, among other effects; they increase line losses and reduce the carrying capacity of power lines. This subject can be investigated deeply for transient, time-varying loads and resonance conditions. This will create new research issues and the implication of intelligent electronic devices (IEDs) in advanced metering infrastructures (AMI) will, easily, help to reach flexible control of real industrial electric systems as well as for transmission, distribution, residential and commercial electrical networks with distributed resources and FACT devices.

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215

5. Power quality indices update In the balanced sinusoidal three-phase situation, the positive component (balanced component at the fundamental frequency) can be considered as a best harmonic signature due to the absence of first and second unbalanced components and the absence of balanced components at harmonic frequencies as well. Moreover, in the general case (unbalance and non sinusoidal conditions), the distinction between the power quality deterioration due to the customer or due to the power system itself is a difficult task. This leads to the conclusion that there is no simple generally valid way for the power system to identify the main contributors to the poor power quality. In our opinion, the efforts of system operators must be focused on the search of the best mitigating procedures of unbalance and harmonics. As well as it is the case, all components of equations (26) and (27) proposed in the paper can be avoided or mitigated except the positive component (balanced component at the fundamental frequency). System operators may, also, be penalized in case of non-compliance to some quality objectives. The penetration of the first and second unbalanced components can be standardized and the negative or zero unbalance factors at the fundamental frequency, traditionally, used in power quality assessment can be extended to harmonic domain. The indices system concerns both voltages and currents. In the following, we will, only, focus on the voltage indices for the sake of brevity. The following factors are customary adopted at the fundamental frequency: 1 CUFfu =

1 V¯ fu 1 V¯ bn

,

1 CUFsu =

1 V¯ su 1 V¯

(37)

bn

1 and CUF 1 are negative and zero complex unbalance factors respectively. where CUFfu su In the general case, new complex indices are proposed for each harmonic order (including fundamental one). The proposed factors are as follows: h V¯ fu

h ICUFfu =

h V¯ bn

,

h ICUFsu =

h V¯ su h V¯

(38)

bn

h and ICUF h are individual first and second harmonic complex unbalance factors respectively. ICUFfu su Another characterization of unbalance in the presence of harmonics can be established by means of the new scalar indices as follows: h NUFfu =

h Vfu 1 Vbn

,

h NUFsu =

 N 

G1 MUFfu

=

m=1

=

m=1

G0 MUFfu

=

3m−2 Vfu

2

3m−1 Vfu

m=1 1 Vbn

3m Vfu

 N

,

G1 MUFsu

m=1

=

2 G2 MUFsu

2

=

 ,

G0 MUFsu

=

3m−2 Vsu

2 (40)

1 Vbn

 N

,

1 Vbn

 N 

(39)

1 Vbn

1 Vbn

 N  G2 MUFfu

h Vsu

m=1

3m−1 Vsu

2

1 Vbn

N

m=1

3m Vsu

(41)

2

1 Vbn

(42)

h and NUF h are normalized first and N corresponds to the higher integer number which gives a higher harmonic order of interest. NUFfu su G G second unbalance factors, respectively. MUFfu1,2,0 and MUFsu1,2,0 are cumulative first and second harmonic unbalance factors, respectively. Cumulative indices must be recommended when voltage or current spectra provide rich harmonic components even if their magnitudes have low values. The tracking’s systems of these factors by power quality monitoring programs can attest for any distortion forms and provide best ways 1 (S 1 the positive apparent power at the fundamental frequency) is used to sizing for mitigating techniques evaluation. The ratio SeN /Sbn bn filter devices. The harmonic sequence components provide a good estimate of filter performance as well as for their control algorithms at all harmonic orders of interest such as recommended for selective/overall active filters and network-wide harmonic control scheme using active filters.

6. Examples Two examples are analyzed, in order to apply the previous study. 6.1. Example 1 The voltage and current phasors in phase reference frame of a three-phase four-wire unbalanced and non-sinusoidal system are listed in Table 1 [17]. Table 2 shows the effective voltages based sequence and phase reference frames. The proposed method gives results close to those given in reference [17] which are based on the phase reference frame.

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O. Boudebbouz et al. / Electric Power Systems Research 117 (2014) 210–218

Table 1 Voltage and current phasors of the studied system. Voltages or currents

Harmonic order 1

3

5

V¯ ah [V] V¯ bh [V] V¯ ch [V]

90∠0◦ 95∠ − 118◦ 98∠124◦

1∠ − 15◦ 0.6∠ − 25◦ 0.8∠ − 5◦

4∠155◦ 4∠260◦ 5∠ − 30◦

I¯ah [V] I¯bh [V] I¯ch [V]

10∠ − 20◦ 8∠ − 130◦ 7∠95◦

3∠ − 75◦ 2∠ − 85◦ 2∠ − 60◦

2∠88◦ 1∠190◦ 3∠ − 50◦

Table 2 Effective Voltage and current based sequence and phase reference frames. Voltages or currents

Phase reference frame. Results given in [17].

Sequence reference frame. The proposed method.

Effective voltage Ve [V] Effective harmonic voltage VeH [V] Effective current Ie [A] Effective harmonic current IeH [A] Effective apparent power Se [KVA] Effective harmonic apparent power SeH [VA]

94.44

94.4588

3.89

4.2543

10.21

10.2115

5.20

5.1611

2.89

2.8936

60.68

65.8706

6.2. Example 2 This example is taken from the reference [14]. The corresponding voltage and current phasors are shown in Table 3. Resulted effective voltages, currents and apparent powers based sequence and phase reference frames are shown in Table 4. 6.3. Indices calculation The new proposed indices are calculated in order to perform the power quality assessment in more convenient way related to unbalance phenomena in the presence of harmonics. The example in [14] is reconsidered for this purpose. Table 5 shows normalized first and second unbalance factors. Table 5 shows precious information about normalized first unbalance factors at all harmonic orders. Their magnitudes exceed one at the 5 can attest of the overestimation of the positive sequence which can be increased fundamental frequency. The relative great value of NUFfu

7 but with noting that the negative to an inacceptable level and further dielectric stress. The same remark can be established for NUFfu sequence would create further losses. The other important information is that of the normalized second unbalance factors of 5th and 7th harmonic orders which exceed one at the fundamental frequency of about 177% and 268.15%, respectively. These indices provide consistent information of the behavior of electrical equipments under unbalance conditions in the presence of h will attest of the parasitic torque for which harmonics. The case of the rotating machines is perhaps the relevant one. For h = 3m − 2, NUFfu

Table 3 Percent harmonic voltage and current phasors base values: Va1 = 271.03 [V]; Ia1 = 99.98 [A]. h

1

3

5

7

9

Vah (%) ◦ ˇah

100 −0.74

10.28 6.76

4.92 142.3

7.44 146.7

8.64 −47.4

Vbh (%) ◦ ˇbh

104.49 −121.2

10.53 6.28

5.79 167.4

8.58 125.2

11.05 −49.19

Vch (%) ◦ ˇch

103.73 121.3

8.69 9.7

4.3 157.7

6.58 136.5

8.22 −47.37

Iah (%) ◦ ϕah

100 −22

68.84 100

34.90 −175

27.85 −65

5.93 48

79.77 99.49

42.30 65.09

45.81 −167.9

40.59 41.89

Ibh (%) ◦ ϕbh

93.49 −120.8

Ich (%)

0

◦

h ˇa,b,c , the voltage phase angles in degree.

0

0

0

0

O. Boudebbouz et al. / Electric Power Systems Research 117 (2014) 210–218

217

Table 4 Effective voltages and currents based sequence and phase reference frames. Voltages or currents

Phase reference frame. Results given in [14].

Sequence reference frame. The proposed method.

Effective voltage Ve [V] Effective harmonic voltage VeH [V] Effective current Ie [A] Effective harmonic current IeH [A] Neutral current In [A] Harmonic neutral current InH [A] Effective apparent power Se [KVA] Effective harmonic apparent power SeH [KVA]

280.25

280.2573

31.68

31.7170

165.08

164.6608

125.41

124.8107

210.66

209.6790

168.90

167.5871

138.79

138.4421

11.92

11.8758

Table 5 Normalized first and second unbalance factors. h

1

3

h NUFfu h NUFsu

0.24%

0.31%

2.7%

0.77%

5

7

9

0.4%

0.56%

1.13%

4.79%

7.24%

0.47%

Table 6 Cumulative first and second harmonic unbalance factors. G1 MUFfu

G2 MUFfu

G0 MUFfu

G1 MUFsu

G2 MUFsu

G0 MUFsu

0.61%

0.4%

1.17%

7.72

4.79%

0.9%

the rotor would be slowed down while for 3m − 1, it will be accelerated. Cumulative indices can, also, be used for torques interaction evaluation in rotating machines. The sequence impact of grouped harmonic components of each subset (G1 , G2 and G0 ) is evaluated by the cumulative indices as shown in Table 6. As stated in [18], the angle of complex unbalance factors at the fundamental frequency (V¯ −1 /V¯ +1 ) has been presented as an important quantity that merits particular attention in analyzing voltage unbalance issues. The same analysis can be extended to harmonic domain by means by of the individual first and second harmonic complex unbalance factors for a better accuracy. 7. Discussions and comments Tables 2 and 4, related to both examples, show that the new formulas give results very similar to those based on the phase reference frame. These findings may be useful for an audience who may be interested in the details of mitigating the polluting components with more accuracy. They, also, provide more information about the harmonic unbalance duality and its effect on power system operations. On the other hand, power quality assessment can be characterized by some new indices. It is, indeed, possible to interpret the unbalance behavior by looking to the individual unbalanced components corresponding to each harmonic order. The phasors of the 1st and the 2nd unbalanced components can be, easily, compared to the balanced components phasors at each harmonic order. The unbalance penetration in power systems can be so quantified by the ratio of the 1st or the 2nd unbalanced components to balanced components like it is customary adopted for the fundamental frequency. Further developments can be done in order to identify scattered components of the instantaneous power based harmonic sequence components. In the deregulation power systems, distributed generators and FACTS devices will raise the performance, the efficiency and the reliability of power systems to higher level. Moreover, the power quality issues must be questioned regarding their non linear behavior. Due to this fact, the smartness of power grids will be reached by providing some sophisticated analysis such as those proposed in the present paper. Moreover, the same analysis will be addressed by the Langella method and it will be presented in the future works, especially, for comparison purposes. 8. Conclusions The paper has presented a more sophisticated formula of effective apparent power. The latter has been derived by means of symmetrical harmonic components. The characterization of the energy flow by the proposed harmonic sequence components can be very suitable in diagnosis tasks and provide convenient ways for the evaluation of the harmonic mitigation and balancing procedures. The power quality monitoring and reporting programs will have very interesting opportunities for developing software in accordance with the proposed harmonic sequence components. So, the smartness of power grids will be supported by consistent factors which are easy to monitor in on-line supervisory systems.

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Acknowledgements This work was supported by the Electric Power System Laboratory “LES” (Laboratoire d’électrotechnique de Skikda) of the University of 20 August 1955, Skikda, Algeria. References [1] P.S. Filipski, Y. Baghzouz, M.D. Cox, Discussion of power definitions contained in the IEEE dictionary, IEEE Trans. Power Deliv. 9 (3) (1994) 1237–1244. [2] L.S. Czarnecki, Misinterpretations of some powers properties of electric circuits, IEEE Trans. Power Deliv. 9 (4) (1994) 1760–1769. [3] L. Cristaldi, A.M. Ferrero, G. Superti-Furga, Current decomposition in asymmetrical, unbalanced three-phase systems under non-sinusoidal conditions, IEEE Trans. Instrum. Meas. 43 (1) (1994) 63–68. [4] J.L. Willems, J.A. Ghijselen, A.E. Emanual, The apparent power concept and the IEEE standard 1459–2000, IEEE Trans. Power Deliv. 20 (2) (2005) 876–884. [5] S. Pajic, A.E. Emanuel, Modern apparent power definitions: theoretical versus practical approach: the general case, IEEE Trans. Power Deliv. 21 (4) (2006) 1787–1792. [6] F. Ghassemi, New concept in AC power theory, IEE Proc. Gener. Transm. Distrib. 147 (6) (2000) 417–424. [7] DIN 40110, Part 2. Multiconductor – circuits, German Standard AC Quantities, Germany, November 2002. [8] IEEE Trial-Use Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Non sinusoidal,. Balanced, or Unbalanced Conditions. IEEE Std 1459-2010 (Revised Version). [9] P. Tenti, J.L. Willems, P. Mattavelli, E. Tedeschi, Generalized symmetrical components for periodic non-sinusoidal three-phase signals, Electr. Power Qual. Util. 13 (2007) 9–15. [10] T. Zheng, E.B. Makram, A.A. Girgis, Evaluating power system unbalance in the presence of harmonics, IEEE Trans. Power Deliv. 18 (2) (2003) 393–397. [11] O. Boudebbouz, K. Zehar, A. Boukadoum, Harmonic signature for diagnosis purposes in unbalanced distribution systems, Arab. J. Sci. Eng. 34 (2B) (2009) 461–476. [12] O. Boudebbouz, A. Boukadoum, S. Leulmi, Harmonic sequence impedances: a tool for the three-phase frequency scan technique, Arab. J. Sci. Eng. 36 (7) (2011) 1277–1286. [13] R. Langella, A. Testa, A.E. Emanuel, Unbalance definition for electrical power systems in the presence of harmonics and interharmonics, IEEE Trans. Instrum. Meas. 61 (10) (2012) 2622–2631. [14] IEEE Trial-Use Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Non sinusoidal Balanced, or Unbalanced Conditions. IEEE Std 1459-2000. Approved September 2002. [15] J. Arrilaga, N.R. Watson, S. Chen, Power System Quality Assessment, John Wiley & Sons, 2000, pp. 23. [16] A.E. Emanuel, Power Definitions and the Physical Mechanism of Power Flow, Wiley, 2010, pp. 189. [17] D.L. Milanez, A.E. Emanuel, The instantaneous space-phasor: a powerful diagnosis tool, IEEE Trans. Instrum. Meas. 52 (3) (2003) 143–148. [18] Y.-J. Wang, Analysis of effects of three-phase voltage unbalance on induction motors with emphasis on the angle of the complex voltage unbalance factor, IEEE Power Eng. Rev. 16 (3) (2001) 270–275. Omar Boudebbouz was born in 1966 in Algeria. He received his a State Engineer Degree in Electrical Engineering from the Badji Mokhtar University, Annaba, Algeria, the Master Degree “Diplôme de Magister” and the Doctor degree in Electric Power System Modeling and simulation from the University of August 20th, 1955, Skikda, Algeria in 1990, 2000 and 2012 respectively. He worked in industry for about 7 years as joined Engineer – Researcher. Currently, he is an Associate Professor Class B at the University of August 20th, 1955, Skikda, Algeria. His research interests include computer simulation of power systems, power system harmonics and optimal operation and control of power systems.

Ahcene Boukadoum was born in 1969 in Skikda, Algeria. He received the Engineer Diploma and, PhD Degree in Electromechanics, in 1996 and 2000, respectively, from State University of Technology of Tashkent Beirnouni, Uzbekistan, in Energy Systems. He is currently a Professor at the University of Skikda and the Director of The Laboratory ‘LES’, Algeria. His main research field is Electrical Machine and power systems Diagnosis.

Salah Leulmi was born in 1951 in Algeria. He received a State Engineer Degree, in electric power systems, from the National Polytechnic School of Algiers in Algeria (1976), a Master Degree of Engineering, in electric power system engineering, from RPI, Troy, NY, USA (1978) and a PhD, in electrical engineering, from ISU, Ames, Iowa, USA (1983). He is the author of around forty publications & communications in journals & proceedings. He was the Head “Director” of the University of August 20th, 1955, Skikda “ex-ENSET”, Algeria. Currently, he is a Professor & President of the Scientific Council of the Science & Engineering Faculty “School” at the same institution. He is, also, a referee of 4 Algerian Journals & some Proceedings & a referee of one overseas Society “WSEAS” for Proceedings & Journals. He is, also actually, the President of the National State Commission of the Equivalency Degrees.